(_b_) If 2a + 3b is a root of x^2 - 6bx - 4a^2 + 9b^2 = 0, find the other root without solving the equation. 7. (_a_) Solve for x: [2x - 3a]^(1/2) + [3x - 2a]^(1/2) = 3[a^(1/2)]. (_b_) Solve for m: 1 - (1)/(2 - m) = 1/(m + 2) + (m - 6)/(4 - m^2). 8. Solve the system: x^2 + 2y^2 = 17; xy - y^2 = 2. 9. Two boats leave simultaneously opposite shores of a river 2-1/4 mi. wide and pass each other in 15 min. The faster boat completes the trip 6-3/4 min. before the other reaches the opposite shore. Find the rates of the boats in miles per hour. 10. Write the sixth term of [x/(2[y^2]^(1/3)) - (y^(1/2))/x]^9 without writing the preceding terms. 11. The sum of the 2d and 20th terms of an A. P. is 10, and their product is 23-47/64. What is the sum of sixteen terms? ~PRINCETON UNIVERSITY~ ALGEBRA A TIME: TWO HOURS Candidates who are at this time taking _both_ Algebra A and Algebra B may omit from Algebra A questions 4, 5, and 6, and from Algebra B questions 1 (_a_), 3, and 4. 1. Simplify (a^3 + a^2b + ab^2)/(a^2 - 3ab - 4b^2) / {(a^2 + 6ab - 7b^2)/(a^2 + 8ab - 9b^2) . (a^3 - b^3)/(a^2 - 7ab + 12b^2)}. 2. (_a_) Divide a^(5/2) + ab^(3/2) + b^(5/2) - 2a^(1/2)b^2 - a^(3/2)b by a^(3/2) - b^(3/2) + a^(1/2)b - ab^(1/2). (_b_) Simplify (1)/(x^(-1) + y^(-1)} . (x^(1/4)y^(1/2))^3 + 1. 3. Factor: (_a_) (x^2 + 3x)^2 - (2x - 6)^2. (_b_) a^2 + ac - 4b^2 - 2bc. 4. Solve 1/(x + 1) - (1)/(x - 1) - (1)/(x - 3) + (1)/(x - 5) = 0. 5. Solve for x and y: mx + ax = my - by, x - y = a + b. 6. The road from A to B is uphill for 5 mi., level for 4 mi., and then downhill for 6 mi. A man walks from B to A in 4 hr.; later he walks halfway from A to B and back again to A in 3 hr. and 55 min.; and later he walks from A to B in 3 hr. and 52 min. What are his rates of walking uphill, downhill, and on the level, if these do not vary? ALGEBRA B 1. Solve (_a_) (x + 1)/(x - 2) + (2x + 1)/(x + 1) + (3x + 3)/(1 - x) = 0. (_b_) [2x + 7]^(1/2) + [3x - 18]^(1/2) - [7x + 1]^(1/2) = 0. (_c_) 6/(x^2 + 2x) = 5 - 2x - x^2. 2. Solve for x and y, checking one solution in each problem: (_a_) 2x + 3y = 1, 6/x + 1/y = 2. (_b_) x^2 = x + y, y^2 = 3y - x. 3. A man arranges to pay a debt of $3600 in 40 monthly payments which form an A. P. After paying 30 of them he still owes 1/3 of his debt. What was his first payment? 4. If 4 quantities are in proportion and the second is a mean proportional between the third and fourth, prove that the third will be a mean prop. between the first and second. 5. In the expansion of [2x + 1/3x]^6 the ratio of the fourth term to the fifth is 2 : 1. Find x. 6. Two men A and B can together do a piece of work in 12 days; B would need 10 days more than A to do the whole work. How many days would it take A alone to do the work? ALGEBRA TO QUADRATICS